A) the probability is 0.123 or 12.3%.
B) the probability is 0.8589 or 85.89%.
C) the normal distribution can be used even though the sample is small because the population the sample is taken from is normally distributed.
Explanation
A) We first find the z-score that corresponds with this value.
Using a z-table (http://www.z-table.com) we find 1.16; this corresponds to a probability of 0.8770. However, This is the probability that a number will be lower than this; since we want the probability that the value is higher, we subtract from 1:
1-0.8770 = 0.123.
B) To find the probability of a sample mean, we use the formula
Looking in our z-table we see that this corresponds with 0.1611. Again, however, this is the area left of this value, which is the probability below; we want the probability above, so we subtract from 1:
1-0.1611 = 0.8589
B) the probability is 0.8589 or 85.89%.
C) the normal distribution can be used even though the sample is small because the population the sample is taken from is normally distributed.
Explanation
A) We first find the z-score that corresponds with this value.
Using a z-table (http://www.z-table.com) we find 1.16; this corresponds to a probability of 0.8770. However, This is the probability that a number will be lower than this; since we want the probability that the value is higher, we subtract from 1:
1-0.8770 = 0.123.
B) To find the probability of a sample mean, we use the formula
Looking in our z-table we see that this corresponds with 0.1611. Again, however, this is the area left of this value, which is the probability below; we want the probability above, so we subtract from 1:
1-0.1611 = 0.8589